Consider the following combinatorial market problem: A seller wishes to sell a set of items to consumers. Each consumer has a valuation function that assigns a non-negative value to every subset of items . The valuation functions can exhibit various combinations of substitutability and complementarity over items; and as standard, valuations are assumed to be monotone ( for any ) and normalized (). Each consumer has a quasi-linear utility function, meaning that her utility for a bundle that she pays for is
. An allocation is a vectorof disjoint bundles of items, where is the bundle allocated to consumer . The social welfare of an allocation is the sum of consumers’ values for their bundles, i.e., . An allocation that maximizes the social welfare is said to be socially efficient.
A classic market design problem is setting prices so that socially efficient outcomes arise in “equilibrium”. Arguably, the most appealing equilibrium notion is that of a Walrasian Equilibrium (WE) [Walras, 1874]. A WE is a pair of allocation and item prices, where each consumer maximizes her utility, and the market clears, namely all items are allocated.111More precisely, unallocated items have price . A WE is a desired outcome, as it is a simple and transparent pricing that clears the market. Moreover, by the “First Welfare Theorem”, every allocation that is part of a WE maximizes the social welfare222Moreover, every allocation that is part of a WE maximizes welfare also over all feasible fractional allocations [Nisan and Segal, 2006]..
Unfortunately, Walrasian equilibria exist only rarely. In particular, they are guaranteed to exist for the class of “gross substitutes” valuations [Kelso Jr and Crawford, 1982], which is a strict subclass of submodular valuations; and in some formal sense, it is a maximal class for which a WE is guaranteed to exist [Gul and Stacchetti, 1999]. Given the appealing properties of a WE, it is not surprising that various approaches and relaxations have been considered in the literature in an attempt to ameliorate the non-existence problem.
The endowment effect.
Recently, Babaioff, Dobzinski and Oren  (henceforth, Babaioff et al. ), proposed to take a behavioral economic perspective, and harness the endowment effect in order to extend market stability and efficiency. The endowment effect, coined by Thaler , posits that consumers tend to inflate the value of the items they own. This phenomenon was later validated by experiments, which realized and quantified the magnitude of the effect in single item settings [Knetsch, 1989; Kahneman et al., 1990].
Exploring this cognitive bias in more complex settings is of great importance. Babaioff et al. propose capturing the endowment effect in combinatorial settings by formulating an endowed valuation function. Given some valuation function , and an endowed set , the endowed valuation function assigns the following real value to every set , referred to as the endowed valuation of with respect to :
where is the endowment effect parameter, and denotes the marginal contribution of given . The idea behind this formulation is that the value of items already owned by the agent () is multiplied by some factor , while the marginal value of the other items () remains intact.
An endowment equilibrium is then a Walrasian equilibrium with respect to the endowed valuations. The main result of Babaioff et al. is that when consumers’ valuations are submodular and , there exists an endowment equilibrium that gives a 2-approximation to the optimal (even fractional) social welfare with respect to the original valuations. They also show that the existence result does not extend to the more general class of XOS valuations. In particular, for every , there exists an instance with XOS valuations that does not admit an endowment equilibrium.
The specific function given in Equation (1) is one way to formulate the endowment effect in combinatorial settings, but certainly not the only one. We actually believe there is no single formulation that fits all scenarios.
In this paper we open up the discussion regarding how to reason about a strict subset of endowed sets, a terrain that, to the best of our knowledge, has not been discussed at all prior to Babaioff et al. . To this end, we provide a new framework that sets the ground for various formulations of the endowment effect based on fundamental behavioral economic principles, and propose a few concrete endowment effects that we find particularly natural from a behavioral point of view.
Our suggestions are by no means exhaustive. On the contrary, we hope that our work will inspire further discussion regarding meaningful endowment effects in combinatorial settings, as well as experimental work that will shed more light on appropriate instantiations for different scenarios.
1.1 A New Framework for the Endowment Effect
Given a valuation , and an endowed set , let denote the endowment valuation with respect to . In this section we introduce the framework we provide for formulating endowment valuations. Our framework is based on two basic principles, described below.
The “loss aversion” principle.
The loss aversion hypothesis is presented as part of prospect theory and is argued to be the source of the endowment effect [Kahneman et al., 1990, 1991; Tversky and Kahneman, 1979]. This hypothesis claims that
People tend to prefer avoiding losses to acquiring equivalent gains.
The loss aversion principle can be formulated as follows:
That is, the loss incurred due to losing a previously-endowed set is greater than the benefit derived from being awarded a set not having been owning before.
The “residual equivalence” principle.
The additional principle, proposed by Babaioff et al. , states that the endowment effect with respect to set should maintain the marginal contribution of items outside of intact. That is, given set , only the value of items in may be subject to the endowment effect. This principle is formulated as follows:
Since , Equation (3) can be rewritten as
That is, for every set , the difference between and is some function of the sets , and . Thus, the endowment valuation of with respect to is given by:
for some function . The function is referred to as the gain function with respect to . It describes the added effect an endowed set has on the consumer’s valuation.
Based on this formulation, Condition (2) becomes a condition solely on the functions ; namely, it requires that for all (see Section 3.1). An endowment effect formulation, or in short: an endowment effect, is then given by a collection of functions for each that satisfy this condition. An endowment environment is given by a vector of endowment effects for the consumers . When clear in the context, we sometimes use the terms endowment effect and endowment environment interchangeably, for example when the endowment effect of all consumers is the same.
In Section 3.3 we define a partial order over endowment effects, which naturally extends to a partial order over endowment environments. We show that an endowment effect that “dominates” another endowment effect based on preserves its stability. More formally, given two endowment environments, , such that , a Walrasian equilibrium with respect to the endowed valuations according to is also a Walrasian equilibrium with respect to the endowed valuations according to (Corollary 3.8).
Specific endowment effects.
The following are four concrete endowment effects within our framework that we find particularly natural from a behavioral point of view.
We discuss each effect through the term — the additional loss incurred upon losing a subset of an endowed set due to the endowment effect. In Definition 3.6 we provide a partial order over all endowment effects, based on this loss.
The identity endowment effect:
It holds that . I.e., it doubles the marginal value of every subset of an endowed set. This is precisely the endowment effect considered by Babaioff et al. for the case of .
The sum of marginals endowment effect: .
It holds that . I.e., upon losing a single item , the additional loss incurred due to the endowment is , identically to the identity effect. extends this rationale in a simple additive manner, representing consumers whose bias is applied to each item separately.
The absolute loss endowment effect: .
It holds that . I.e., demonstrates a bias that is more “loss averse” than identity (Definition 3.6 formalizes this “loss aversion”), where a consumer amplifies the loss of a subset of an endowed set by “forgetting” the fact that remains at the consumer’s hand. For subadditive valuations, this is a larger loss (see Proposition A.4 in Appendix A.2).
The all or nothing endowment effect:
It holds that for any strict subset . Intuitively, the consumer “regains” rationality upon losing any previously endowed items. This instance represents a bias that is stimulated only for the bundle as a whole.
1.2 Existence of Equilibria and Welfare Approximation
In this section we present our existence and approximation results. All of our approximation results hold with respect to the optimal welfare according to the original valuations, and even with respect to the optimal fractional allocation.333 Note that, by the First Welfare Theorem, an endowment equilibrium always gives the optimal welfare with respect to the endowed valuations.
Recall that Babaioff et al. prove that every market with submodular consumers admits an -endowment equilibrium that gives a -approximation welfare guarantee. In Corollary 4.13 we show that the same proof technique can be used to prove that the exact same result holds also with respect to the weaker endowment effect . (We show in Proposition 3.9 that for submodular consumers, .)
For the larger class of XOS consumers, Babaioff et al. show that an endowment equilibrium may not exist even with respect to an endowment effect for an arbitrarily large . This negative result may lead one to conclude that while the endowment effect improves stability for submodular valuations, XOS markets may remain unstable even with respect to endowed valuations. However, we show that this negative result is an artifact of the specific formulation chosen by the authors. As established in the following theorem, the stronger endowment effect leads to a sweeping existence and approximation result for markets with XOS valuations.
Theorem [Thm. 4.1] Every market with XOS consumers admits an -endowment equilibrium that gives a -approximation welfare guarantee.
Moreover, we show a natural better response dynamics for endowed valuations that converges to an -endowment equilibrium. The dynamics is a variant of the dynamics considered by Fu et al. ; Christodoulou et al. .
The theorem above shows that a stronger endowment effect enables extending the equilibrium existence (and approximation) result from submodular valuataions to XOS valuations. Can this result be extended further? In Section 4.4, we explore more general valuations. First, we provide an endowment effect that inflates the value of a set linearly in its size, and show that there always exists an -endowment equilibrium. In contrast, we show that for any endowment effect with a “reasonable” inflation (specifically, up to ), an endowment equilibrium may not exist for subadditive valuations.
1.3 The Power of Bundling
We next study the power of bundling in settings with endowed valuations. A bundling is a partition of the set of items into disjoint bundles. A competitive bundling equilibrium (CBE) [Dobzinski et al., 2015] is a bundling and a Walrasian equilibrium in the market induced by (i.e., the market where are the indivisible items). It is easy to see that a CBE always exists (say, bundle all items together, and assign the grand bundle to the highest value consumer for a price of the second highest value). However, in contrast to WE, CBEs do not have any meaningful welfare guarantees in general. Recent literature studied competitive bundling equilibra (and variants thereof) with good welfare guarantees [Feldman and Lucier, 2014; Feldman et al., 2016; Dobzinski et al., 2015].
In this paper we introduce the notion of -endowment CBE. An -endowment CBE is a CBE with respect to the endowed valuations. We provide welfare guarantees for -endowment CBEs, for any endowment effect satisfying a mild assumption that for all . Endowment effects satisfying this assumption are said to be significant. For example, one can easily verify that , and are all significant with respect to all valuations, while is not necessarily significant.
Interestingly, bundling leads to a sweeping positive result when combined with significant endowment effects:
Theorem [Thm. 5.4] Every market (with arbitrary valuations) admits an -endowment CBE with optimal social welfare, for every significant endowment environment .
For example, a direct corollary of the last theorem is that every market, no matter how complex valuations are, admits an endowment CBE with optimal social welfare, when considering the endowment effect formulation considered by Babaioff et al. . We note that this result cannot be extended to all endowment effects within our framework. In particular, for endowment effects such that for some it holds that for all , there are instances that admit no endowment CBE with optimal welfare, already for XOS valuations (Proposition 5.5). For this subclass of endowment effects, we provide approximation lower bounds as a function of the parameter , for different classes of valuations (including XOS, subadditive, and arbitrary; see Section 5).
Babaioff et al. showed computational barriers towards computing -endowment equilibria, and raised the following question (recall that denotes the endowment effect that multiplies each gain function by ):
Are there allocations that can be both efficiently computed and paired with item prices that form an -endowment equilibrium for a small value of ?
As shown in Babaioff et al. , every -endowment equilibrium gives approximation to the optimal social welfare. In contrast, when turning to endowment CBEs, no meaningful approximation can be guaranteed in general (see Section 3.2). Therefore, in settings with bundling, the question should be reformulated as follows:
Are there approximately optimal allocations that can be both efficiently computed and paired with bundle prices, that form an -endowment CBE for some natural endowment effect ?
We provide the following positive results.
Theorem [Thm. 5.7] For submodular valuations, and a significant endowment environment , there exists a polynomial algorithm that given an allocation , produces an allocation and bundle prices , so that is an -endowment CBE with at least as much social welfare as . The algorithm invokes a polynomial number of value queries.
Theorem [Thm. 5.8] For general valuations, and a significant endowment environment , there exists a polynomial algorithm that given an allocation , produces an allocation and bundle prices , so that is an -endowment CBE with at least as much social welfare as . The algorithm invokes a polynomial number of demand queries
This effectively reduces the problem of finding significant endowment CBEs to the problem of algorithmic welfare maximization.
1.4 Comparison to Related Work
Our work builds upon the recent work by Babaioff et al.  that proposed the first formulation for the endowment effect in combinatorial auctions. They show that every market with submodular valuations admits an -endowment equilibrium that gives at least half of the optimal social welfare.
Other relaxations of WE have been considered in the literature in an attempt to ameliorate the non-existence problem of WE, and achieve approximate stability and efficiency for more general valuation classes than gross substitutes.
Fu et al.  considered a relaxed notion of WE, termed conditional equilibrium. A conditional equilibrium is a pair of an allocation and item prices satisfying individual rationality, and such that no consumer wishes to expand their allocation, but disposing of items is not allowed. They showed that every conditional equilibrium has at least half of the optimal welfare. Moreover, every market with XOS valuations admits a conditional equilibrium, which can be reached via a “flexible ascent auction”, an algorithm proposed by Christodoulou et al. .444Our results imply that their approximation guarantee applies also with respect to the optimal fractional social welfare. (To the best of our knowledge, this was not previously known.
A different relaxation of WE was considered by Feldman et al. , where the utility maximization condition is preserved, but market clearance is relaxed (i.e., items with positive prices may be unsold). Using this notion, an equilibrium always exists (say, price all items at some prohibitively large price), but such equilibria carry no approximation guarantees. For this notion it is shown that even for simple markets with two submodular consumers, the social welfare approximation guarantee cannot be better than .
Our results on endowment CBE (Section 5) should be compared with previous notions of bundling equilibria [Feldman and Lucier, 2014; Feldman et al., 2016; Dobzinski et al., 2015]. In these settings, the market designer first partitions the set of items into indivisible bundles (these are the indivisible items in the induced market), and assigns prices to these bundles instead of the original items, and a CBE is a Walrasian equilibrium in the induced market.
Dobzinski et al.  showed that every market (with arbitrary valuations) admits a CBE that gives approximation guarantee of . Moreover, given an optimal allocation, a CBE with such approximation can be computed in polynomial time. Furthermore, they provide a polynomial time algorithm that computes a CBE with a approximation guarantee.
This should be compared to Theorems 5.4 and 5.8 in this paper. Theorem 5.4 shows that for a wide variety of endowment effects (including the one considered by Babaioff et al. ), there always exists an endowment CBE that gives the optimal welfare. Theorem 5.8 shows that for a wide variety of endowment effects (including that of Babaioff et al. ), given an arbitrary allocation , one can compute, in polynomial time, an endowment CBE with (weakly) higher welfare than . Thus, the problem of computing nearly-efficient endowment CBEs is effectively reduced to the pure algorithmic problem of welfare approximation — a problem with vast literature (e.g., [Dobzinski et al., 2005; Lehmann et al., 2006; Dobzinski and Schapira, 2006; Feige and Vondrak, 2006; Feige, 2009; Feige and Izsak, 2013; Chakrabarty and Goel, 2010]).
A different notion of bundling equilibria was considered by Feldman et al. . This notion is a relaxed version of CBE, where some bundles (with positive prices) may remain unsold. Under this notion, for arbitrary valuations, given an arbitrary allocation , one can compute, in polynomial time, an equilibrium with welfare at least half of the welfare of .
All the notions above consider a concise set of bundles, a price for each bundle, and an additive pricing over sets of bundles. More general forms of bundle pricing, including non-linear and non-anonymous pricing, lead to welfare-maximizing results, but are highly impractical (in particular, they use an exponential number of prices) [Bikhchandani and Ostroy, 2002; Parkes and Ungar, 2000; Ausubel and Milgrom, 2000; Lahaie and Parkes, 2009; Sun and Yang, 2014].
1.5 Paper Organization
Section 2 presents some preliminaries on Walrasian equilibria, valuation classes and query models. The endowment effect framework is described in Section 3.1, followed by Section 3.2 that establishes efficiency guarantees for endowment equilibria, and Section 3.3 which describes the partial order over endowment effects. In Section 4 we provide existence results for endowment equilibria, and in Section 5 we introduce the notion of endowment-Competitive Bundling Equilibrium (CBE), and provide existence, approximation and computational guarantees with respect to this notion. All missing proofs are deferred to the appendix.
Consider a market with a set of items and consumers. Each consumer has a valuation function that assigns a real value for every subset of items . As standard, assume that valuations are normalized; i.e., , and monotone (free-disposal), i.e., for any , . An allocation is a partition of to disjoint bundles where bundle is allocated to consumer .
In this work we measure the quality of an allocation by its social welfare . An item pricing is a vector where is the price of item . Given an allocation and item pricing , consumer ’s quasi-linear utility is
Given a price vector and a set , we use .
(Walrasian Equilibrium) A pair of an allocation and a price vector is a Walrasian Equilibrium (WE) if:
Utility maximization: Every consumer receives an allocation that maximizes her utility given the item prices, i.e., for every consumer and bundle .
Market clearance: All items are allocated, i.e., .
We define the classes of valuation functions considered in this paper, from least to most general, except for unit demand valuations and budget additive valuations which have no containment relation.
Unit demand: if there exist values , so that .
Budget additive: if there exist values , so that .
Submodular: if for any it holds that .
Fractionally subadditive (XOS): if there exist vectors so that for any it holds that .
Subadditive: if for any it holds that .
Value and demand queries.
The representation of combinatorial valuation functions is exponential in the parameters of the problem. A standard computational model in this setting is an oracle access. We consider value and demand queries.
A value query for valuation receives a set as input, and returns .
A demand query for valuation receives a price vector as input, and returns a set that maximizes .
3 Endowment effect
3.1 Endowment Effect Framework
In the introduction, we present two principles that underlie the endowment effect, namely the loss aversion principle and the residual equivalence principle. The loss aversion principle states that:
and the residual equivalence principle states that
The endowed valuation satisfies the residual equivalence principle if and only if
For simplicity of presentation, we also assume that the gain functions are normalized; i.e., for all and , it holds that . This implies that the endowed valuations are also normalized; i.e., . Our results can be generalized to non-normalized gain functions.
Based on this characterization, the following definition follows.
An endowment effect is a collection of gain functions for each , such that for all . Given an endowment effect , a valuation function , and an endowed set , the endowment valuation with respect to is given by
For simplicity, when the endowment effect is clear in the context, we write instead of . An endowment environment is given by a vector of endowment effects for the consumers .
We are now ready to define the notion of endowment equilibrium.
For an instance and endowment environment , a pair of an allocation and a price vector forms an -endowment equilibrium, if is a Walrasian equilibrium with respect to ; i.e.,
Utility maximization: Every consumer receives an allocation that maximizes her endowed utility given the item prices, i.e., for every consumer and bundle ,
Market clearance: All items are allocated, i.e., .
We abuse notation and use both for endowment effect and endowment environment when all consumers are subject to the same endowment effect.
3.2 Efficiency Guarantees for Endowment Equilibria
Given an endowment environment , we are interested both in the existence and the social welfare of
-endowment equilibria. It is well known that Walrasian equilibria are related to the following linear program relaxation for combinatorial auctions (known as theconfiguration LP, see e.g., [Bikhchandani and Mamer, 1997]):
For each : .
For each : .
For each :
The existence of a Walrasian equilibrium turns out to be closely related to the integrality gap of the configuration LP:
[Nisan and Segal, 2006] An instance admits a Walrasian Equilibrium if and only if the integrality gap of the configuration LP is . Moreover, an integral allocation has payments such that is a Walrasian Equilibrium if and only if is an optimal solution to the LP.
The following proposition gives an approximation guarantee for every endowment equilibrium, as a function of the gain functions. This is a natural generalization of [Babaioff et al., 2018, Corollary 3.7]. Note that an additional requirement is that the gain functions are non-negative.
Given an instance , let be the value of the optimal fractional welfare. For an endowment effect , where for all , if is an -endowment equilibrium, then the allocation has welfare guarantee of at least
where for all .
Since is an -endowment equilibrium, by Theorem 3.3, for any optimal fractional solution of the LP w.r.t. the valuations it holds that
where the first inequality is by optimality and the second is by non-negativity of the gain functions. The proof follows by multiplying both sides by and rearranging. ∎
An immediate corollary is the following:
If is an -endowment equilibrium for instance , and for all it holds that , then the social welfare of is a -approximation to the optimal fractional welfare.
Corollary 3.5 implies a -approximation guarantee for submodular consumers with endowment effect, or more generally, with any of the endowment effects listed in the following section.
3.3 Partial order over endowment effects
Recall that an endowment effect is specified by a set of gain functions for every set .
We next define a partial order over the space of endowment effects.
Fix a valuation function , and two endowment effects with respect to .
Given a set , we write if for all , .
We write (and say that is dominated by , or dominates ) if for all it holds that .
We next show that an endowment effect always preserves the endowment equilibria of endowment effects dominated by it (Corrolary 3.8). To prove this, we use the following lemma.
For an instance , and an endowment environment , let be an -endowment equilibrium. For any consumer , and endowment effect , if , then is also an -endowment equilibrium.
Let , and let . The pair is an -endowment equilibrium, therefore for every it holds that
Since , by Definition 3.6, the last inequality still holds when is replaced by . I.e.,
Rearranging, we conclude that:
i.e., that It follows that maximizes consumer ’s utility, as desired. Individual rationality follows by the fact that endowed valuations are normalized. ∎
By applying Lemma 3.7 iteratively for each consumer , we get the following:
Suppose is an -endowment equilibrium with respect to instance , and let be such that for every , . It holds that is also an -endowment equilibrium.
Recall the specific endowment effects considered in Section 1 (in what follows, it is assumed that , by the definition of ).
The sum of marginals endowment effect
The identity endowment effect:
The absolute loss endowment effect
The all or nothing endowment effect
For submodular valuations, the endowment effect dominates the endowment effect .
For every submodular valuation , it holds that .
Fix a set , and let and . For all we need to show that . By the additivity of , it follows that . Therefore, it remains to show that .
Rename the items in by , and let denote the set of items . It holds that
where the inequality holds by submodularity, and the last equality holds by definition of . ∎
By Corollary 3.8, it follows that every -endowment equilibrium is also an -endowment equilibrium.
4 Existence of endowment equilibrium
In this section we establish existence and efficiency guarantees of endowment equilibria in combinatorial auctions.
The main theorem in this section is that for every instance with XOS valuations, there exists an -endowment equilibrium. Moreover, there exists an algorithm that given an initial allocation, finds an -endowment equilibrium with at least as much welfare.
There exists an algorithm that for any instance of XOS valuations, and any initial allocation , returns an -endowment equilibrium such that .
In Section 4.1 we show the connections between endowment equilibrium and conditional equilibrium. In Section 4.2 we provide the proof of Theorem 4.1. In Section 4.3 we show that for every instance with submodular valuations, there exists an -endowment equilibrium (recall that . We conclude in Section 4.4, showing that our existence results cannot be extended to subadditive valuations. In particular, for every endowment effect in our framework, there exists an instance with subadditive valuations that does not admit an endowment equilibrium.
4.1 Endowment Equilibrium and Conditional Equilibrium
Our analysis draws upon the relation between an endowment equilibrium and a conditional equilibrium [Fu et al., 2012]. The definition of a conditional equilibrium follows.
[Fu et al., 2012] For an instance , a pair of allocation and item pricing is a conditional equilibrium if for all ,
Outward stability: For every ,
We first introduce the notion of inward stability. A set is inward stable if the consumer never wishes to dispose of any subset of items from (possibly by replacing them with a different set of items). Formally:
Given a consumer with valuation , and item pricing , a set is inward stable w.r.t. and if for every it holds that .
This means that if is inward stable for consumer , then there exists a utility-maximizing set of items for consumer that contains .
In general, endowment and conditional equilibria are incomparable notions. The following proposition shows that any endowment equilibrium that is also individually rational with respect to the original valuations is a conditional equilibrium.
For any instance , if a pair of allocation and item prices is an -endowment equilibrium, and for all consumers it holds that , then is a conditional equilibrium.
Individual rationality is given. It remains to show outward stability. For any consumer with endowment effect , since is an endowment equilibrium, it holds that for every , i.e.,
is linear so as required, ∎
In the other direction, for a conditional equilibrium to be an endowment equilibrium, it needs to be inward stable with respect to the endowed valuations. In the following lemma we give a sufficient condition for inward stability.
Given a consumer with valuation , an endowment effect , and item pricing , if a set satisfies for all , then is inward stable with respect to and .
Fix any . By monotonicity of we have that:
It is given that Combining the two inequalities above implies that as required. ∎
The following proposition shows that an allocation and prices that are both inward stable with respect to the endowed valuations, and outward stable, form an endowment equilibrium.
For any instance , if the pair of allocation and item prices is a conditional equilibrium, and the endowment environment is such that for every consumer , the gain function satisfies for all , then is an -endowment equilibrium.
Fix a consumer and a set . It is given in the proposition that the conditions of Lemma 4.5 on with , and hold, therefore, Since is a conditional equilibrium, it holds that . It follows that
therefore, consumer is utility maximizing. Finally, note that individual rationality follows by considering the case . ∎
4.2 Proof of Main Result (Theorem 4.1)
In this section We show that the -endowment effect leads to strong existence and efficiency guarantees in combinatorial markets with XOS valuations. In particular, we show that: (1) every market with XOS valuations admits an -endowment equilibrium, (2) an optimal allocation can always be paired with prices such that forms an -endowment equilibrium, and (3) an -endowment equilibrium can be reached using a natural dynamics that is a modified version of the “flexible ascent auction” presented by Fu et al.  (given here as Algorithm 1). Recall also that by Corollary 3.5, every allocation that is part of an -endowment equilibrium gives at least half of the welfare of the optimal (even fractional) allocation.
We begin by recalling the definition of supporting prices [Dobzinski et al., 2005]. Given a valuation and a set , the prices are supporting prices for if and for every , . It was shown (e.g., in [Dobzinski et al., 2005]) that a valuation is XOS if and only if for all there exist supporting prices for .
The following lemma shows that for XOS valuations, the condition of Lemma 4.5 holds with respect to the endowment effect , and a set of supporting prices.
Fix a consumer with an XOS valuation . If are supporting prices w.r.t. and , then satisfies for all .
Observe that by definition of supporting prices, it holds that for any . By definition of , we have that . Rearranging, we conclude that
as required. ∎
The above lemma has an immediate implication. Recall that in [Fu et al., 2012] it was shown that in an instance of XOS valuations, for any welfare-maximizing allocation , if one sets the prices of items in each to be the supporting prices with respect to and , then is a conditional equilibrium. Combining the last observation with Lemma 4.7 and Proposition 4.6, we conclude the following:
Fix an instance of XOS valuations. For any welfare-maximizing allocation there exist prices such that is an -endowment equilibrium.
We now show that given a starting allocation , one can run a modified version of the “flexible ascent auction” from [Fu et al., 2012], that results in an -endowment equilibrium with at least as much welfare as . Moreover, also satisfies individual rationality with respect to the original valuations, which, by Proposition 3.4 and Proposition 4.4 implies that is a conditional equilibrium and that approximates the optimal fractional welfare OPT up to .
The main difference of Algorithm 1 compared to the flexible ascent auction is that in the end of every iteration all the prices may change, not only the ones demanded in the current iteration. Specifically, given that at some iteration consumer , who was previously allocated , is now allocated for some , then for all such that , the prices of may change, so that the prices are supporting prices with respect to every consumer, and thus inward stability is maintained. This property implies that restricting attention to deviations that take the form of extending the current allocation is without loss.
The following lemma shows that the dynamics in Algorithm 1 are better-response dynamics.
Let and be the allocation and price vector in the beginning of some iteration in Algorithm 1. For the chosen consumer , and her corresponding set , it holds that . I.e., consumer performs a beneficial deviation.
Note that in the end of every iteration, all prices are adjusted to be supporting prices for every consumer. This implies, by Lemmas 4.5 and 4.7, that the allocation of every consumer is inward stable.
At each iteration of Algorithm 1, the social welfare strictly increases
At the beginning of every iteration, for every consumer , the prices are supporting prices with respect to